\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x+f x^2)^2} \, dx\) [143]

Optimal result

Integrand size = 27, antiderivative size = 730 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\frac {\left (2 b-\frac {c e}{f}\right ) \sqrt {a+b x+c x^2}}{e^2-4 d f}+\frac {2 c x \sqrt {a+b x+c x^2}}{e^2-4 d f}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{f^2}-\frac {\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \left (e-\sqrt {e^2-4 d f}\right )+2 f \left (c d f (b e-4 a f)-f^2 \left (2 b^2 d-3 a b e+4 a^2 f\right )-2 c^2 d \left (e^2-4 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {2} f^2 \left (e^2-4 d f\right )^{3/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\left ((c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) \left (e+\sqrt {e^2-4 d f}\right )+2 f \left (c d f (b e-4 a f)-f^2 \left (2 b^2 d-3 a b e+4 a^2 f\right )-2 c^2 d \left (e^2-4 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {2} f^2 \left (e^2-4 d f\right )^{3/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \] Output:

(2*b-c*e/f)*(c*x^2+b*x+a)^(1/2)/(-4*d*f+e^2)+2*c*x*(c*x^2+b*x+a)^(1/2)/(-4 
*d*f+e^2)-(2*f*x+e)*(c*x^2+b*x+a)^(3/2)/(-4*d*f+e^2)/(f*x^2+e*x+d)+c^(3/2) 
*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/f^2-1/4*((-b*f+c*e)*(f 
*(-2*a*f+b*e)+2*c*(-5*d*f+e^2))*(e-(-4*d*f+e^2)^(1/2))+2*f*(c*d*f*(-4*a*f+ 
b*e)-f^2*(4*a^2*f-3*a*b*e+2*b^2*d)-2*c^2*d*(-4*d*f+e^2)))*arctanh(1/4*(4*a 
*f-b*(e-(-4*d*f+e^2)^(1/2))+2*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c 
*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x 
+a)^(1/2))*2^(1/2)/f^2/(-4*d*f+e^2)^(3/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b 
*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+1/4*((-b*f+c*e)*(f*(-2*a*f+b*e)+2*c*(-5* 
d*f+e^2))*(e+(-4*d*f+e^2)^(1/2))+2*f*(c*d*f*(-4*a*f+b*e)-f^2*(4*a^2*f-3*a* 
b*e+2*b^2*d)-2*c^2*d*(-4*d*f+e^2)))*arctanh(1/4*(4*a*f-b*(e+(-4*d*f+e^2)^( 
1/2))+2*(b*f-c*(e+(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a 
*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1/2)/f^2 
/(-4*d*f+e^2)^(3/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^( 
1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 5.92 (sec) , antiderivative size = 2854, normalized size of antiderivative = 3.91 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2)^2,x]
 

Output:

((-2*f*Sqrt[a + x*(b + c*x)]*(c*e^2*x - b*f*(2*d + e*x) + c*d*(e - 2*f*x) 
+ a*f*(e + 2*f*x)))/((e^2 - 4*d*f)*(d + x*(e + f*x))) + 4*c^(3/2)*ArcTanh[ 
(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])] + (2*RootSum[c^2*d - b*c*e 
 + b^2*f + 2*Sqrt[a]*c*e*#1 - 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 + b*e*#1^2 + 4 
*a*f*#1^2 - 2*Sqrt[a]*e*#1^3 + d*#1^4 & , (-(c^3*d^4*Log[x]) + b*c^2*d^3*e 
*Log[x] + 4*a*c^2*d^3*f*Log[x] - 6*a*b^2*d^2*f^2*Log[x] - 7*a^2*c*d^2*f^2* 
Log[x] + 9*a^2*b*d*e*f^2*Log[x] - 4*a^3*e^2*f^2*Log[x] + 4*a^3*d*f^3*Log[x 
] + c^3*d^4*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - b*c^2*d^3*e*Log 
[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 4*a*c^2*d^3*f*Log[-Sqrt[a] + S 
qrt[a + b*x + c*x^2] - x*#1] + 6*a*b^2*d^2*f^2*Log[-Sqrt[a] + Sqrt[a + b*x 
 + c*x^2] - x*#1] + 7*a^2*c*d^2*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - 
 x*#1] - 9*a^2*b*d*e*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + 4* 
a^3*e^2*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 4*a^3*d*f^3*Log 
[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 2*Sqrt[a]*c^2*d^3*e*Log[x]*#1 
+ 4*a^(3/2)*b*d^2*f^2*Log[x]*#1 - 2*a^(5/2)*d*e*f^2*Log[x]*#1 + 2*Sqrt[a]* 
c^2*d^3*e*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 - 4*a^(3/2)*b*d^ 
2*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + 2*a^(5/2)*d*e*f^2* 
Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + c^2*d^4*Log[x]*#1^2 - a^ 
2*d^2*f^2*Log[x]*#1^2 - c^2*d^4*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*# 
1]*#1^2 + a^2*d^2*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2...
 

Rubi [A] (warning: unable to verify)

Time = 1.70 (sec) , antiderivative size = 716, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1302, 27, 2138, 27, 2143, 27, 1092, 219, 1365, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1302

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left (8 c f x^2+2 (3 c e+b f) x+3 b e-4 a f\right )}{2 \left (f x^2+e x+d\right )}dx}{e^2-4 d f}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left (8 c f x^2+2 (3 c e+b f) x+3 b e-4 a f\right )}{f x^2+e x+d}dx}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 2138

\(\displaystyle \frac {-\frac {\int \frac {2 \left (-2 f \left (e^2-4 d f\right ) x^2 c^3+f \left (2 d f b^2-e (c d+3 a f) b+4 a f (c d+a f)\right ) c-f \left (2 d e c^2+2 a e f c+b \left (e^2-10 d f\right ) c+b f (b e-2 a f)\right ) x c\right )}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{2 c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {-2 f \left (e^2-4 d f\right ) x^2 c^3+f \left (2 d f b^2-e (c d+3 a f) b+4 a f (c d+a f)\right ) c-f \left (2 d e c^2+2 a e f c+b \left (e^2-10 d f\right ) c+b f (b e-2 a f)\right ) x c}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 2143

\(\displaystyle \frac {-\frac {\frac {\int \frac {c f \left (2 d \left (e^2-4 d f\right ) c^2+f \left (2 d f b^2-e (c d+3 a f) b+4 a f (c d+a f)\right )+(c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) x\right )}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}-2 c^3 \left (e^2-4 d f\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {c \int \frac {2 d \left (e^2-4 d f\right ) c^2+f \left (2 d f b^2-e (c d+3 a f) b+4 a f (c d+a f)\right )+(c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx-2 c^3 \left (e^2-4 d f\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {-\frac {c \int \frac {2 d \left (e^2-4 d f\right ) c^2+f \left (2 d f b^2-e (c d+3 a f) b+4 a f (c d+a f)\right )+(c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx-4 c^3 \left (e^2-4 d f\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {c \int \frac {2 d \left (e^2-4 d f\right ) c^2+f \left (2 d f b^2-e (c d+3 a f) b+4 a f (c d+a f)\right )+(c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx-2 c^{5/2} \left (e^2-4 d f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {-\frac {c \left (\frac {\left (\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}-\frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}\right )-2 c^{5/2} \left (e^2-4 d f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {c \left (\frac {2 \left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}-\frac {2 \left (\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}\right )-2 c^{5/2} \left (e^2-4 d f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {c \left (\frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\left (\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)+2 c \left (e^2-5 d f\right )\right )-2 f \left (f \left (-b e (3 a f+c d)+4 a f (a f+c d)+2 b^2 d f\right )+2 c^2 d \left (e^2-4 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )-2 c^{5/2} \left (e^2-4 d f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c f^2}-\frac {2 \sqrt {a+b x+c x^2} (-2 b f+c e-2 c f x)}{f}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2)^2,x]
 

Output:

-(((e + 2*f*x)*(a + b*x + c*x^2)^(3/2))/((e^2 - 4*d*f)*(d + e*x + f*x^2))) 
 + ((-2*(c*e - 2*b*f - 2*c*f*x)*Sqrt[a + b*x + c*x^2])/f - (-2*c^(5/2)*(e^ 
2 - 4*d*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])] + c*(((( 
c*e - b*f)*(f*(b*e - 2*a*f) + 2*c*(e^2 - 5*d*f))*(e - Sqrt[e^2 - 4*d*f]) - 
 2*f*(2*c^2*d*(e^2 - 4*d*f) + f*(2*b^2*d*f + 4*a*f*(c*d + a*f) - b*e*(c*d 
+ 3*a*f))))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - S 
qrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - 
(c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 
- 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4 
*d*f]]) - (((c*e - b*f)*(f*(b*e - 2*a*f) + 2*c*(e^2 - 5*d*f))*(e + Sqrt[e^ 
2 - 4*d*f]) - 2*f*(2*c^2*d*(e^2 - 4*d*f) + f*(2*b^2*d*f + 4*a*f*(c*d + a*f 
) - b*e*(c*d + 3*a*f))))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b 
*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f 
 + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt 
[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f) 
*Sqrt[e^2 - 4*d*f]])))/(c*f^2))/(2*(e^2 - 4*d*f))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1302

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x 
_)^2)^(q_), x_Symbol] :> Simp[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e 
*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) 
   Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p 
+ 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ 
[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !IGtQ[q, 0]
 

rule 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f 
*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 
] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
 

rule 2138

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. 
)*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] 
, C = Coeff[Px, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + 
 q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q 
 + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 
 1)*(2*p + 2*q + 3))   Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Si 
mp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + 
(p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2* 
A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - 
 B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 
 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*( 
c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q 
 + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C* 
d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, 
e, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2* 
p + 2*q + 3, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]
 

rule 2143

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ 
.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C 
 = Coeff[Px, x, 2]}, Simp[C/c   Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 
1/c   Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x 
^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(7855\) vs. \(2(664)=1328\).

Time = 3.03 (sec) , antiderivative size = 7856, normalized size of antiderivative = 10.76

method	result	size
default	\(\text {Expression too large to display}\)	\(7856\)

Input:

int((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+b*x+a)**(3/2)/(f*x**2+e*x+d)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e x + d\right )}^{2}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(3/2)/(f*x^2 + e*x + d)^2, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (f\,x^2+e\,x+d\right )}^2} \,d x \] Input:

int((a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2)^2,x)
                                                                                    
                                                                                    
 

Output:

int((a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^2} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{\left (f \,x^{2}+e x +d \right )^{2}}d x \] Input:

int((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x)
 

Output:

int((c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d)^2,x)